I would like to acknowledge that Professors Lin and Toomre

of MIT are also interested in the problem of spiral structure,

and that I have benefited from discussions with them as well

as their students.

*Kalnajs 1963, p.13*

**3.1 Working hypothesis and semi-empirical theory**

In hindsight, considering the crucial influence that the

Lin & Shu (1964)
paper had on the thinking of astronomers, it is

only regretful that Lin did not decide (with or without me) to

publish even earlier, because he certainly had all the physical

ideas contained in our paper well before 1964.

*Shu 2001*

While Toomre, Hunter and Kalnajs had already presented their first results
in the dynamics of flat galaxies, Lin still kept on thinking over the spiral
problem. ^{63} Astronomers
in Princeton had
convinced him that, despite Chandrasekhar's criticism of Lindblad's
theories, ^{64} the idea
itself of a
long-lived, shape-preserving spiral pattern is consistent with Hubble's
classification system that relates spiral features with a galaxy's
morphological type, its steady characteristic, thus suggesting that the
spirals are steady as well. This view reminded Lin of wave modes in fluid
flows that he had been studying for years back.
^{65} On purely heuristic
grounds, discrete spiral modes
seemed to him very reasonable as the natural result of wave evolution,
and, if so, the patterns released might be associated with *slowly
growing* or *neutral* modes. Lin raised
this premise to the rank of working hypothesis, and around it as the nucleus
he set to develop a *semi-empirical* theory.
^{66} It was seen to follow
best the "urgent assignment from
the astronomers [...] to make some specific calculations" and "to
demonstrate the possibility of the existence of quasi-stationary spiral
modes from the theoretical point of view [...] with understanding of the
dynamical mechanisms relegated to a secondary and even tertiary position"
(*Lin*). ^{67}^{,}^{68}

"The conclusion in the working hypothesis is

not proved or deuce, but supported by an accumulation of theoretical analysis and empirical data. The adoption of this working hypothesis is a very important step in the development of a theory of spiral structure. It means that the authors are committed to back it up with the comparison of subsequent predictions with observational data." (Lin)

The coauthor to share Lin's fame and commitment was his student Frank Shu
(Shu 1964)
^{69} who
"found it remarkable that a scientist trained as a professional
mathematician would place higher priority on empirical facts than deductive
reasoning" and believed that "it was this broad-mindedness and clear
vision that gave Lin a considerable advantage over his many competitors of
the period" (*Shu*).
^{70} The Lin and Shu paper
"On the spiral structure of disk galaxies"
(Lin & Shu 1964,
hereinafter LS64), in which "they first demonstrated the
plausibility of a purely gravitational theory for density waves by a
continuum treatment"
(Lin & Shu 1966,
p.459),
appeared in August 1964. ^{71}

The paper considered small non-axisymmetric disturbances to a razor-thin cold disk and found for them, through the governing hydrodynamic and Poisson equations, wave-like solutions of the type

(12) |

each specified by its eigenfunction
(*r*)
and a
pair of eigenvalues and
*m*. For further advancement,
the WKBJ-method was applied. It is valid for the case of phase
*S*(*r*) varying with radius much faster than amplitude
*A*(*r*),
which features the *tightly wrapped* spirals, ones of small
pitch angle between the circumferential tangent and the tangent
to the constant-phase line

(13) |

Depending on the sign of a radial-wavenumber function *k*(*r*) = -
*S* /
*r*, the
spirals are trailing (*k* > 0) or leading (*k* < 0)
(Fig.8).
With *A*(*r*) expanded in a series over a small parameter
tan*i* = *m* / *kr* (*i* being the pitch angle), the
problem is solved to the
lowest, *i*-independent order neglecting the azimuthal force
component of spiral gravity. In this case, both leading and
trailing arms act as just rings, so that the ensuing dispersion
relation

(14) |

substantially repeats Toomre's equation (5) for radial
oscillations. Importantly, relation (14) is valid for
Re{^{2}}
1. This restricts the
radial span of the WKBJ solutions, and in the neutral case
Im{}= 0 they
gain the territory between the Lindblad resonances determined by
Eqn (11) and equating the angular speed of an *m* -armed spiral
pattern to a combination

(15) |

with the minus/plus sign discriminating, respectively, between the ILR and OLR. The two-armed spirals thus seem preferred as best covering an entire disk (Fig.9).

Such was the mathematical basis of the original Lin-Shu density-wave theory,
called elementary by its authors any later (e.g.
Bertin & Lin 1996,
p.229).
It treated wave quantities
_{p},
, and
*m* as free
parameters burdened with no dynamical imposition, which made the theory so
comfortable in imitating spiral grand designs by means of the curves
*r*() given by

(16) |

and obtained through the integration of expressions (13)
and (14). Sure, the results of this procedure were controvertible,
already because the *fast*-growing waves - exactly those
examined in LS64 - ruled out the proclaimed
quasi-stationarity. ^{72}
But the authors hoped that random motions, excluded from their
analysis, would in fact stave off disk instability as
definitively as to impose a state of near-stability open for
*slowly* growing modes until a small but finite amplitude.

Toomre (1964)
had reflected already on such a state of *Q*
1 as
settling *once* all over the disk-like stellar Galaxy, but yet he
found it stable* still*,
at least in our solar region. As a counterpoise, Lin with Shu diagnosed
instability for another region, at about
*r*_{0} = 4 - 5 kpc from the
center. With that, they pictured "a galactic disk, which is in part stable
and in part unstable" and suggested "the possibility of a balance
resulting in a neutral density wave extending over the *whole* disk
and having a
scale of the order of (but smaller than) the distance between the stable and
unstable regions" (LS64, p. 651). It was this "suggestion of the
possibility" that summarized Lin's early reflections and made his basic
working hypothesis originally sound as a statement that

"the total stellar population, which has various degrees of velocity dispersion, forms a

quasi-stationary spiral structurein space of the general nature discussed above" (LS64, p.651).

As we can see, this statement hinges almost entirely on the opinion that,
for our galactic disk to be equally stable at that *r*_{0},
the velocity dispersion must there exceed
*c*_{r, min}
80 ± 10 km/s, which
cannot be the case, else "a considerable number of stars with high radial
velocities would reach our neighborhood from the interior part of the
Galaxy, contrary to observational evidence" (LS64, p.651). But was this
opinion (the authors never repeated it) strong enough? First, it meant an
inconceivable situation when some *massive* portion of a stellar
galaxy remains *unstable* during all the period of formation in it
of a global quasi-*steady* pattern.
Secondly, and most important for astronomers, it had - already in 1964 -
grave objections to the fact that the largest epicyclic deflection of the
Lin-Shu "high radial velocity stars" from their `home' radius
*r*_{0} = 4 - 5 kpc, equaled to
*r*
*r*_{0}
*c*_{r} / *V*_{0} 2^{1/2}, was
in frames of Schmidt's model (cited in LS64) 1 - 1.5 kpc only - too
little to let those stars even come close from *r*_{0}, if
not reach us. We
find that the original QSSS hypothesis of Lin and Shu, called nowadays "a
preliminary formulation" *only*
(Bertin & Lin 1996,
p.80), rested on a rather weak basis, both dynamical and empirical.

Very interesting in LS64 is the authors' notice on what had made their work
get to print so urgently. A passage following their opening discussion of
"at least two possible types of spiral theories", one of which "is to
associate every spiral arm with a *given body of matter*" and the
other "is to regard the spiral
structure as a [quasi-steady] *wave pattern*", reads:

"Toomre tends to favor the first of the possibilities described above. In his point of view, the material clumping is periodically destroyed by differential rotation and regenerated by gravitational instability.

^{73}[...] The present authors favor the second point of view [...] Since A. Toomre's (1964) point of view has been published, it seems desirable to publish our point of view even though the work is not yet as complete as the present writers would wish to have it." (LS64, p.646)

This puzzles. Although it is true that from about 1962 onward Toomre
suspected - much as Lynden-Bell had already done in his thesis two years
earlier, as it turned out - that at least the more ragged-looking spiral
structures result primarily from recurrent gravitational instabilities in
the plainly dissipative gas layer of a galaxy (*Toomre*), there was
no explicit
discussion of any such suspicions in T64 as actually published. One cannot
help but think that this accentuated mention of
`Toomre (1964)'
was more than just a mistaken reference, that actually it betrayed the
influence that at least the cited paper had on Lin.

Shu: "Here, I can only speculate, because certainly my foresight then was not as sharply developed as Lin's. Nor was I privy to the developing estrangement between him and Alar Toomre. [...] Lin had been thinking about the problem of spiral structure nonstop since the Princeton conference in 1961. But he had a world-renowned reputation to protect and therefore was loathe to publish anything hasty before he had worked out his ideas mathematically to his satisfaction. [...] Lin (and later, I) felt strongly that spiral structure was, in essence, a normal mode. But by all the standards of what was then known, a normal mode could not be spiral (unless it grew ridiculously fast). Nevertheless, Lin felt sure that one should not do the naive thing of superimposing equal trailing and leading parts when the wave frequency is (nearly) real. And he probably wanted to discover the reason why before publishing anything. Alar's 1964 paper triggered him into premature action". (Shu)

Lin: "The urgency in my submittal of our paper was to present adifferentperspective, not to fight for priority". "After reviewing the paper again, I think I could not have done much better or even any better". (Lin)

One way or another, we see that by 1964 Lin indeed had had several thoughts
and feelings about spiral modes, and he was eager about gaining power to his
perspective. At that, he knew of a growing optimism with shearing or
evolving density waves ^{74}
and, as well, of the
parallel wave-mode interest at Harvard. The T64 paper
^{75} , apart from its
engagements
on disk stability, did mention Kalnajs' advancing efforts and, still more
glaringly, it also mentioned and already *discussed* Lin's yet
unpublished solutions. ^{76}
This must have put Lin in a
position to urgently patent his views, albeit makeshift in argument for want
of better mathematics, and in so doing he rather awkwardly exhibited the
opponents' preoccupations as an alternative already placed on record.

^{63} Lin's basic themes still were in
hydrodynamics (e.g.,
Benney & Lin 1962;
Reid & Lin 1963).
Back.

^{64} That criticism
(Chandrasekhar 1942)
concerned only the
asymptotic-spiral theory, and it was itself not flawless as attached to
confusing empirical data of the 1920's - 30's.
Back.

^{65} "I have been
thinking of modes ever since I learned about the fine points of the Hubble
classification". (*Lin*)
Back.

^{66} "I adopted the empirical approach
because of my close contacts with the observers (and with Lo Woltjer). Now
that I have thought over the situation some more, I think I should admit
that it is probably true that my past long-standing experience in the
studies of hydrodynamic instability did (as you hinted) play a role in my
thinking (although I was not conscious of it). But more important, I also
feel (upon reflection) that the reason I adopted the empirical approach is
really the natural consequence of my past education. My undergraduate
education was in physics (at Tsinghua University of China, where all the
major professors in Physics had doctorate degrees from English speaking
universities such as Harvard, Caltech, Chicago and Cambridge), with all the
pleasant memories of doing the experiments with precision and the
satisfaction of having the data checked against theory. My graduate
education was primarily at Caltech where I studied under Theodore von
Karman. It is also there that I took a course from Fritz Zwicky who first
identified the regular spiral structure in the Population II objects of the
Whirlpool M51". (*Lin*)
Back.

^{67} "Despite of my decades of experience
with instability of
shear flows, I did not bring these matters into the presentation of the 1964
paper, but commented only vaguely about instability. [...] There was no
shortage of theoretical astronomers who understood the mechanisms perhaps
better than I did; e.g. Lo Woltjer and Donald Lynden-Bell and perhaps even
Peter Goldreich (even at that point). Goldreich turned out be the most
successful leader in the understanding of the density waves in the context
of planetary rings". (*Lin*)
Back.

^{68} "In hindsight, I think Lin's
judgment was accurate considering how quick people were to attack his point
of view with proofs of `antispiral theorems' and the like shortly after the
publication of LS64". (*Shu*)
Back.

^{69} "All the original ideas were
C.C. Lin's, and my
original contributions were mainly to check the equations that he wrote down
and posed as problems. (I did find a way to derive the asymptotic relation
between density and potential by attacking the Poisson integral directly,
but even there I initially blundered in not realizing the necessity of an
absolute value on the radial wavenumber. The final derivation presented in
the appendix of LS64 is due to Lin). I did considerable reading, however, on
the astronomical side and may have contributed some ideas concerning how OB
stars form and die in spiral arms. (This was the beginning of my lifelong
interest in star formation.) Lin was indeed quite generous to include me as
a coauthor on LS64, and I will always be grateful for his guidance and
support of a young (I was 19 at the time) undergraduate
student". (*Shu*)
Back.

^{70} "Lin undoubtedly encouraged many of
his younger
colleagues - like Alar Toomre - to think about the problem of spiral
structure. I can only imagine that Lin's treatment of people then much more
junior than himself was equally as generous as his treatment of myself.
Certainly, he must have discussed with Alar Toomre (and later Chris Hunter)
his ideas about this problem. Toomre's early papers on the subject
acknowledge this debt of introduction and inspiration. Why then did those
early papers not carry Lin's name as a coauthor? I do not know, nor would I
dare to probe (by asking either Lin or Toomre) for fear of opening old
wounds that are best left closed". (*Shu*)

One way or another, no alliance
was formed between Lin and Toomre. They "diverged in emphasis from the very
beginning", so that "there were discussions, but no real collaboration"
(*Lin*). As in agreement with this Toomre recalls that back again
at MIT in
spring 1963 he did decline Lin's "astonishing suggestion to write some such
paper jointly, since he himself had contributed almost nothing very
concretely to my gravitational (in)stability insights, and yet also since I
likewise felt I had added next to nothing to his own spiral-wave hopes"
(*Toomre*).
Back.

^{71} That the historical Lin & Shu
article was referred to as
`Lin's (1963)
preprint' by
Layzer (1964) and as
`Lin (1964)' by
Toomre (1964) and
Kalnajs (1965)
as it was about to appear in the fall of 1964
speaks of its urgently extended coauthorship as Lin's last moment decision
(so striking for a well-motivated and ambitious scientist).

Anyway, the
Lindblad (1964)
paper, also considering quasi-stationary circulation and the
resulting spirals in differentially rotating galaxies, appeared half a year
*prior* to Lin's patent. The authors had neither contacts nor fresh
news on each
other's most parallel work, and hardly could have it. "There was no
justification to trouble B. Lindblad with a novice being converted, Lin
explains. I was waiting for a definitive new prediction before writing to
him. Even then I would have done it through P.O. Lindblad for several
obvious reasons. Unfortunately, by the time our result came out (IAU
Symposium No 31) [see Sect.3.2] he already
passed away" (*Lin*). Even less
probable was any contact-making step from the other side. "About that time
[fall of 1964] my father was on a trip around the world caused by the
inauguration of the Parkes telescope in Australia, P.O. Lindblad recalls. On
his way home he passed through the US [...] but he brought no news about
density wave theories. [...] I think my father was aware of the
existence of the LS64 paper but had not had the time to penetrate it. I know
that he was happy to learn from Whitney Shane, who visited us around the
beginning of June 1965, that his work on spiral structure had been more and
more appreciated recently". (*P.O. Lindblad*)
Back.

^{72} To soundly fit the empirical 2-3
kpc local-arm spacing in the Milky Way, LS64 chose a combination
of angular speed
_{p} = 10
km/s/kpc and growth rate
= 50
km/s/kpc (!) for their tentative two-armed spiral.
Back.

^{73} "The prevalent thinking among the
other prominent theorists of the time -
and this included Alar Toomre - was that spiral structure was a chaotic and
regenerative phenomenon - `shearing bits and pieces', as Alar later put it
in one of his papers". (*Shu*)
Back.

^{74} Goldreich and Lynden-Bell in England and
Julian and Toomre at MIT set to work on this by 1964.
Back.

^{75} The revised
version of T64 was submitted in January 1964.
Back.

^{76} Toomre concluded that "whatever
differences there may exist between the
shorter axisymmetric and non-axisymmetric disturbances, these must in
essence be due only to the circumstance of *differential* rotation"
(T64, p.1223). In
Lin's hands, in contrast, this `circumstance' still allowed the dispersion
relation (14) for non-axisymmetric waves to be rather close to its
axisymmetric analog (5), although the waves stood as steady-mode solutions
of the WKBJ type. Yet, as well, the governing equations admitted an
"altogether different family of approximate non-axisymmetric solutions"
(T64, p.1223), with the radial wavenumber proportional to the disk shear
rate *A*(Oort's constant), and growing with time,
*k*_{r}
*At*. This meant that a spiral disturbance of the leading form
(*t* < 0) unwrapped,
started trailing, and then wrapped tighter and tighter (*t* >
0). Thus the
point was that, on the one hand, differential rotation continuously deforms
even the tightly-wound spiral waves of this sort, whereas, on the other
hand, these "should probably be regarded as particular superpositions of
Lin's solutions" (T64, p.1223). This discordance was thought to be removed
by a fuller analysis beyond the WKBJ-limit.
Back.